Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι → ι be given.
Assume H0: ∀ x3 . x3 ∈ x0 ⟶ x1 x3 = x2 x3.
Claim L1: ∀ x3 . x3 ∈ x0 ⟶ x1 x3 ⊆ x2 x3
Let x3 of type ι be given.
Assume H1: x3 ∈ x0.
Let x4 of type ι be given.
Assume H2: x4 ∈ x1 x3.
Apply H0 with
x3,
λ x5 x6 . x4 ∈ x5 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Claim L2: ∀ x3 . x3 ∈ x0 ⟶ x2 x3 ⊆ x1 x3
Let x3 of type ι be given.
Assume H2: x3 ∈ x0.
Let x4 of type ι be given.
Assume H3: x4 ∈ x2 x3.
Apply H0 with
x3,
λ x5 x6 . x4 ∈ x6 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Apply set_ext with
Pi x0 (λ x3 . x1 x3),
Pi x0 (λ x3 . x2 x3) leaving 2 subgoals.
Apply Pi_cod_mon with
x0,
x1,
x2.
The subproof is completed by applying L1.
Apply Pi_cod_mon with
x0,
x2,
x1.
The subproof is completed by applying L2.